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This chapter demonstrates that the experimental observable in diffraction analysis is the intensity of each diffraction spot which provides only the amplitude of the corresponding structure factor and not its phase — the fundamental phase problem in crystallography. To obtain an image of the molecule forming a crystal we need to calculate a Fourier transform, which requires that we know both the amplitude and phase of each structure factor. Nevertheless, very important information can be derived by calculating a ‘phase-less’ Fourier transform of the intensities alone, which is known as the Patterson function. Although this function is inherently more complex than an electron density map since it displays all inter-atomic vectors, certain sections of the Patterson function, known as Harker sections, can yield information on the positions of the most electron-rich atoms within the crystal. The Patterson function is exploited in most methods of the solving the phase problem for proteins and simple rules for the interpretation of a Patterson function are derived.

This chapter covers methods of structure solution which are used when molecular replacement is impossible, usually due to the lack of a suitable search model. These methods require that additional atoms are introduced into the protein, either by soaking the crystal or by special expression methods. These atoms are chosen to perturb the diffraction pattern of the protein which, firstly, allows the positions of the added atoms to be determined and, secondly, allows the phases of the protein molecule to be calculated. Methods for analysing the data from such derivatives are covered in detail as are methods for locating the additive atoms and refining their positions, many of which rely on the Patterson function. The chapter then introduces the theory of calculating the phases of the protein and treatment of the associated errors. The important role which anomalous scattering, most notably that of selenomethionine, plays in phasing is described in detail along with the associated theory. Density modification methods for improving the accuracy of the resulting phases are then covered.

This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the curves of various functions. It demonstrates the essential principles of diffraction by determining the Fourier transforms of regularly repeating patterns which can be represented mathematically by Dirac delta functions — the very important concept of Fourier space (or reciprocal space) follows from this discussion. This section leads into a description of another highly important mathematical concept, the convolution. Convolutions allow two functions to be combined and provide an extremely elegant mathematical description of the crystalline state as well as an insight into one of crystallography's most important structure-solving tools, the Patterson function.

Starting from a definition of the relationship between a crystal structure and its Patterson function, this chapter considers the role of the Patterson function in the particular context of powder diffraction. The limitations of powder data and their effects upon the appearance of derived Patterson maps are discussed, as are ‘conventional’ and ‘non-conventional’ (e.g., maximum entropy) methods for improving their interpretability. The utility of the Patterson function in the context of improving intensity estimates for overlapping reflections is considered, as is the use of automated procedures for the location of atomic positions based on Patterson map superposition.

This chapter provides an overview of the methods of structure solution of incommensurately modulated crystals and composite crystals. Assuming the periodic average structure is known, it is shown that modulation functions can often be determined by trial and error, employing structure refinements starting with randomly chosen but small values for the structural parameters. The presentation of systematic methods of structure determination includes Patterson function methods, direct methods, and the method of charge flipping.

A diffraction pattern is the (forward) Fourier transform of a crystal structure, obtained physically in an experiment and mathematically from a known or model structure (providing both amplitudes and phases). The (reverse) Fourier transform of a diffraction pattern is an image of the electron density of the structure, unachievable physically, and mathematically, possible only if estimates are available for the missing reflection phases for combination with the observed amplitudes. This chapter considers computing aspects of the required calculations. Variants on the reverse Fourier transform arise from the use of different coefficients instead of the observed amplitudes: squared amplitudes, with no phases, give the Patterson function; ‘normalised’ amplitudes give an E-map in direct methods; differences between observed and calculated amplitudes give difference electron density maps, with applications at various stages of structure determination; weighted amplitudes emphasize or suppress particular features. The concepts are illustrated with a one-dimensional example based on a real structure.

This chapter focuses on symmetry in reciprocal space. It reviews the definitions of reciprocal space and dual (reciprocal) basis vectors (from Chapter 6), and presents the new concept of the reciprocal lattice. It introduces in a very general way the concept of Fourier transform of lattice functions, and explains how these can be naturally assigned to the ‘nodes’ of the reciprocal lattice, thereby producing a pattern that has no translational invariance but has nevertheless either the point-group symmetry of the crystal class or a higher symmetry (the Laue class). The chapter also describes in some detail the concept of extinction (or reflection) conditions, and explains how these conditions relate to the translational and roto-translational symmetry in the real space. The Patterson function, an important concept related to crystal structure solution, is also presented.

This chapter starts with the Fourier series and Fourier transforms, and the representation of periodic functions. It then looks at Fourier analysis in crystallography. It details electron density, the derivation of relations between Fourier coefficients and structure factors, the X-ray resolution of a crystal structure, and the structural analysis of crystals and molecules (trial and error methods, the Patterson function, interpretation of Patterson maps, heavy atom and isomorphous replacement techniques, direct methods, and charge flipping). Finally, it provides an analysis of the Fraunhofer diffraction pattern from a grating and the Abbe theory of image formation.

The methods described in the Chapters 7 and 8 are the only ones practicable when the structure under study is completely unknown. But as soon as a few structures were known, they suggested that related problems might be tackled in a more direct way. For example, if I know the molecular structure of horse oxyhaemoglobin, surely it can help me in the study of horse deoxyhaemoglobin crystals. Also it should help me to tackle the crystal structure of human oxyhaemoglobin. And, by the way, shouldn’t the first protein crystal structure at atomic resolution, that of sperm-whale myoglobin, help me with all of these? Well-developed techniques are now available to apply these ideas. They will be presented first for the simplest case. Suppose I know accurately the structure of a protein crystal form, A, and I can use this crystal structure to define a molecule M. This requires decisions about the boundaries that identify a molecule in A. Now I have crystallized the protein in a different crystal form, X, the diffraction intensities of which I have measured. How can the crystal structure of X be determined? From what I know of protein structure, it is fair to assume that the two molecular structures are very similar. M can be used as a model for the structure which may be in X. There is a very simple case where the crystal symmetry is unchanged and the unit cells are virtually the same. Such cases may be studied by electron-density difference maps, similar to those described for the assignment of heavy atom positions in Chapter 7. This is quite common, for example, in the case of single-site amino-acid substitutions in a protein, and the difference map may show the electron-density differences at the substituted amino acid, while the absence of significant density in the rest of the map could confirm that negligible changes had occurred in the rest of the structure. In the more important case, the crystal forms A and X crystallize differently, and the space-group symmetry is probably different. It is important to begin by determining the symmetry and cell dimensions of the X crystals.

Procedures to determine the phases of the structure factors, by isomorphous replacement, by anomalous scattering, or by molecular replacement, were described in the Chapters 7–9. Using one or more of these methods, phases are generated which allow an electron-density map to be calculated, at a resolution to which the phases are thought to be reliable. In many cases this electron density can be confidently interpreted in terms of atomic positions. But this is not always the case. Quite often, the procedures so far described offer a tantalizing puzzle map, with some features which I think I can interpret, but raising many questions. Before devoting effort to interpreting an unsatisfactory electron-density map, a number of procedures are available, which might make a striking improvement. Perhaps the most important strategy is to seek out more isomorphous and anomalous scattering derivatives. Before doing that, there are other possibilities which may improve an electron-density map without any more experimental data. These methods are known collectively as density modification. The first group of methods exploits features of the electron density which result from the packing of molecules into a crystal. Macromolecular crystals composed of rigid molecules have voids between the molecules filled with disordered solvent, often including the precipitants used in the crystallization process. These solvent regions present featureless density between the structured density of the macromolecules. A high-quality electron-density map will show these featureless regions clearly. In a map of poorer quality, the voids between molecules may be clearly defined, but far from featureless. This provides a method to improve the map. Although some solvent molecules are immobilized on the surface of the macromolecule, those further from the surface are in a disordered liquid-like state which presents a uniform density. Except in very small proteins, the majority of solvent is disordered. If such uniform solvent regions can be recognized, they allow surfaces to be defined which separate solvent regions from protein regions. Two procedures are described below. It has become almost a matter of routine to use one or both of these methods.